Properties of Relations


















Relations on a set

The relations in this section are defined on a set.

(The domain and the codomain are the same set.)


















Reflexive

What's a Reflexive relation?

	The relation contains (x,x) for every x in the domain.

	Every member of the domain is related to itself.

Determine whether each relation is Reflexive or not.

	The 'less-than' relation defined of the set of integers.

	The 'less-than-or-equal' relation defined of the set of integers.

	R = {(1,1),(1,2),(2,1),(2,2),(3,4),(4,1),(4,4)} defined on {1,2,3,4}

	S = {(1,1),(1,2),(2,1)}                         defined on {1,2,3,4}


















Classwork
You may work with a partner.

Determine whether each relation is Reflexive or not.

	The 'equal' relation defined of the set of integers.

	The 'not-equal' relation defined of the set of integers.

	P = {(2,1),(3,1),(3,2),(4,1),(4,2),(4,3)} defined on {1,2,3,4}

	T = {(3,4)}                               defined on {1,2,3,4}


















Reflexive Matrix and Graph

What's true about the matrix of a reflexive relation?

	The main diagonal is all ones.

What's true about the graph of a reflexive relation?

	Every node has a loop.


















Symmetric

What's a Symmetric relation?

	If the relation contains (x,y),
	the relation must contain (y,x).

What's an Antisymmetric relation?

	If the relation contains (x,y),
	the relation must not contain (y,x),
	unless x and y are the same item.

Determine whether each relation is Symmetric, Antisymmetric, both, or neither.

	The 'less-than' relation defined of the set of integers.

	The 'less-than-or-equal' relation defined of the set of integers.

	R = {(1,1),(1,2),(2,1),(2,2),(3,4),(4,1),(4,4)} defined on {1,2,3,4}

	S = {(1,1),(1,2),(2,1)}                         defined on {1,2,3,4}


















Classwork
You may work with a partner.

Determine whether each relation is Symmetric, Antisymmetric, both, or neither.

	The 'equal' relation defined of the set of integers.

	The 'not-equal' relation defined of the set of integers.

	P = {(2,1),(3,1),(3,2),(4,1),(4,2),(4,3)} defined on {1,2,3,4}

	T = {(3,4)}                               defined on {1,2,3,4}


















Symmetric Matrix and Graph

What's true about the matrix of a Symmetric relation?

	Same value in symmetric position about the main diagonal.

What's true about the matrix of an Antisymmetric relation?

	Opposite value in symmetric position (or both zeros).

What's true about the graph of a Symmetric relation?

	Every edge is bidirectional.

What's true about the graph of an Antisymmetric relation?

	No edge is bidirectional.


















Transitive

What's a Transitive relation?

	If the relation contains (x,y) and (y,z),
	the relation must contain (x,z).

Determine whether each relation is Transitive or not.

	The 'less-than' relation defined of the set of integers.

	The 'less-than-or-equal' relation defined of the set of integers.

	R = {(1,1),(1,2),(2,1),(2,2),(3,4),(4,1),(4,4)} defined on {1,2,3,4}

	S = {(1,1),(1,2),(2,1)}                         defined on {1,2,3,4}


















Classwork
You may work with a partner.

Determine whether each relation is Transitive or not.

	The 'equal' relation defined of the set of integers.

	The 'not-equal' relation defined of the set of integers.

	P = {(2,1),(3,1),(3,2),(4,1),(4,2),(4,3)} defined on {1,2,3,4}

	T = {(3,4)}                               defined on {1,2,3,4}


















List the properties each relation possesses.
All relations are defined on the set S = { 1, 2, 3, 4 }.

A = { (1,1),(1,2),(2,1),(2,2),(3,3),(3,4),(4,3),(4,4) }

B = { (1,1),(1,2),(1,3),(1,4),(2,2),(2,4),(3,3),(4,4) }